Church's thesis (constructive mathematics)

In constructive mathematics, Church's thesis ${\displaystyle {\mathrm {CT} }}$ is the principle stating that all total functions are computable functions.

The similarly named Church–Turing thesis states that every effectively calculable function is a computable function, thus collapsing the former notion into the latter. ${\displaystyle {\mathrm {CT} }}$ is stronger in the sense that with it every function is computable. The constructivist principle is however also given, in different theories and incarnations, as a fully formal axiom. The formalization depends on "function" and "computable" of the theory at hand. A common context is recursion theory as established since the 1930's.

Adopting ${\displaystyle {\mathrm {CT} }}$ as a principle, then for a predicate of the form of a family of existence claims (e.g. ${\displaystyle \exists !y.\varphi (x,y)}$ below) that is proven not to be validated for all ${\displaystyle x}$ in a computable manner, the contrapositive of the axiom implies that this is then not validated by any total function (i.e. no mapping corresponding to ${\displaystyle x\mapsto y}$). It thus collapses the possible scope of the notion of function compared to the underlying theory, restricting it to the defined notion of computable function. In turn, the axiom is incompatible with systems that validate total functional value associations and evaluations that are also proven not to be computable. More concretely, it affects ones proof calculus in a way that it makes provable the negations of some common classically provable propositions.

For example, Peano arithmetic ${\displaystyle {\mathsf {PA}}}$ is such a system. Instead of it, one may consider the constructive theory of Heyting arithmetic ${\displaystyle {\mathsf {HA}}}$ with the thesis in its first-order formulation ${\displaystyle {\mathrm {CT} }_{0}}$ as an additional axiom, concerning associations between natural numbers. This theory disproves some universally closed variants of instances of the principle of the excluded middle. It is in this way that the axiom is shown incompatible with ${\displaystyle {\mathsf {PA}}}$. However, ${\displaystyle {\mathsf {HA}}}$ is equiconsistent with both ${\displaystyle {\mathsf {PA}}}$ as well as with the theory given by ${\displaystyle {\mathsf {HA}}}$ plus ${\displaystyle {\mathrm {CT} }_{0}}$. That is, adding either the law of the excluded middle or Church's thesis does not make Heyting arithmetic inconsistent, but adding both does.